EF of the same size and shape and placed at an infinite distance from them, by the straight canal CD of incompressible fluid. Let the three plates be all parallel to each other and be placed so that CD shall pass through their centers and be perpendicular to their planes, and let the plates be overcharged. The quantity of redundant fluid in each of the plates AB and ab will be to that in EF as the repulsion of the plate ab on the canal cD to the sum of the repulsions on cD and fD (cf being taken equal to cC'), supposing that the redundant fluid in all three plates is disposed in the same manner.

For first, as the plates AB and ab are at an infinite distance from any other over or undercharged body, the repulsion of AB on the canal Cc in one direction must be equal to that of ab on it in the contrary, and therefore the redundant fluid in AB must be equal to that in ab.

Secondly, the sum of the repulsions of AB and ab on the canal cD must be equal to that of EF on it in the contrary direction, as otherwise some fluid must flow from ab to EF or from EF to ab. But as all three plates are of the same size, and the fluid in them is disposed in the same manner, the repulsions of EF and ab on cD will be to each other as the quantity of redundant fluid in them, and therefore the quantity of redundant fluid in ab will be to that in EF as the repulsion of ab on CD to the sum of the repulsions of AB and ab on it, that is, as the repulsion of ab on CD to the sum of its repulsions on ƒD and cD, for the repulsion of AB on CD is equal to the repulsion of ab on ƒD*.

142] COR. I. If the fluid in these plates is disposed in the same manner as in Prop. XXIX. the quantity of redundant fluid in each of the plates AB and ab will be to that in EF as

AC (p+) to AC (p + 1) + p (Ac − Cc) +

AC2
24c

For by Lemma X. the repulsion of a given quantity of fluid spread uniformly over ab on the column cD; the repulsion of the same fluid on cf; the repulsion of the same quantity of fluid collected in the circumference of the plate ab on the column cD; and the repulsion of the same fluid on cf are to each other as ac; ac+cf −af; and and therefore the whole repulsion of the plate ab on cD is to its repulsion on cf as

ac

ac

ac

2 2 2af'

ac

ac ac2

and therefore the repulsion of ab on cD is to the sum of its repulsions on CD and ƒD as

143] COR. II. Therefore if all the redundant fluid in the plates is spread uniformly, the redundant fluid in each of the plates AB and ab will be to that in EF as AC: AC+ Ac - Cc, and if it is all collected in the circumference, as AC: AC +

A C
Ac

144] COR. III. By Prop. XXIV. it appears that the redundant fluid in the plate AB or ab will bear the same proportion to that in EF though they communicate with EF by separate canals, and whether the canals by which they communicate with it are straight or crooked, or in whatever direction EF is placed in respect of them, provided the situation of AB and ab in respect of each other remains the same. Only it must be observed that if the fluid in the plates is not disposed so as to be in equilibrio, as will most likely be the case if it is disposed as in the two preceding corollaries, it is necessary that the canals should meet them in their centers, for if the fluid in a plate is not in equilibrio, its repulsion on a canal of infinite length will not be the same in whatever part the canal meets it, as it will if the fluid in the plate is in equilibrio.

145] LEMMA XII. Fig. 2. Let BA be an infinitely slender cylindric column of uniform matter infinitely continued beyond A: the

Fig. 2.

K

B

A

1

repulsion of a particle of matter K on this column in the direction BA is proportional to or may be represented by B, supposing the size of the particle and [the] base of the column to be given.

KB'

For draw KC perpendicular to AB continued, and let the point B flow towards C, the fluxion of the repulsion of K on the column equals - CB CB - KB

the fluent of which,

KB2

1 KB'

is nothing when KB

is infinite.

146] LEMMA XIII.

Suppose now KC to represent an infinitely slender cylindric column of uniform matter: the repulsion of KC on the infinite column BA is to the repulsion of the same quantity of matter

collected in the point C on the same column as the nat. log. of

KC + KB

CB

For the repulsion of all the matter therein, when collected at C,

147] LEMMA XIV. The repulsion of CK on a particle at B, in the

CK direction CB, is proportional to KB CB' and the size of the particle B to be given.

supposing the base of CK

For supposing CK to flow, the fluxion of its repulsion on B in the

CK CB direction CB is proportional to X KB KB'

and is nothing when CK is nothing.

148] LEMMA XV. Fig. 3. bases are GEF and HMN and

the fluent of which is

CK KBX CB'

Let GEFHMN be a cylinder whose whose axis is CK. Let the convex

Fig. 3.

피

K

N

BLO

W

surface of this cylinder be uniformly coated with matter, and let GC be small in respect of CK. Let GA be a diameter of the base produced, and D any point therein. The repulsion of the convex surface of the cylinder on the point D in the direction CD is very nearly the same as if all the matter therein was collected in the axis CK and spread uniformly therein.

For let MED and med be two planes infinitely near to each other, parallel to CK and passing through D, and cutting the convex surface in ME and NF and in me and nf, which will consequently be right lines equal to each other and perpendicular to ED; and draw CP perpendicular to ED.

The repulsion of NnfF on D in the direction CD is proportional to Ffx FN PD

FD × ND

X and that of Mme E is proportional to
CD'

Ff
FD

Eex EM PD

X

EDxMD CD'

Ee

But Ff is to Ee as FD to ED, therefore and are each equal

ED

But the repulsion of the same quantity of matter collected in CK is

proportional to

1

1

(Ff+ Ee) × CK 2
2CD

X

KD'

and, as CG is small in respect

2 * KD

therefore the sum of the

of CK, + differs very little from ND MD repulsions of MmeE and NnfF is very nearly the same as if all the matter in them was collected in CK, and consequently the repulsion of the whole convex surface of the cylinder will be very nearly the sam as if all the matter in it was collected in CK.

149] COR. Therefore if BA represents an infinitely thin cylindric column of uniform matter infinitely extended beyond A, the repulsion of the convex surface of the cylinder thereon in the direction BA is very

1

* As neither MD nor ND differ from KD by so much as CB, it is plain that 1 2 + cannot differ from in so great a proportion as that of BC to KD, but MD ND KD

in reality it does not differ from it in so great a ratio as that of CB to KD2, but as it is not material being so exact, I shall omit the demonstration. See A. 1.

[From MS. "A. 1"] Demonstration of note at bottom of page 8, CB=r, CP=b, PF=d, PD=a, CR2+ CD2 = e2,

ND CR2+a2 - 2ad+d2 = e2 - b2 - 2ad + d2 = e2 - f2 - 2ad

nearly the same as if all the matter therein was collected in CK, and therefore is to the repulsion of the same quantity of matter collected CK+ KB CK in the point C thereon very nearly as nat. log. to CB CB'

that

In like manner the repulsion on

the infinite column DA is to the repulsion of the same quantity of

matter collected in C very nearly as nat. log.

CK+ KD CK

CD

to

CD'

150] PROP. XXXI. Fig. 3. Let the cylinder GEFKMN be connected to the globe W, whose diameter is equal to GB and whose distance from it is infinite, by a canal TR of incompressible fluid of any shape, and meeting the cylinder in any part, and let them be overcharged the quantity of redundant fluid in the cylinder will be to that in the globe in a less ratio than that of CK to nat. log.

CK 2CB

CK
CB'

2CK

, CB

and in

a greater ratio than that of to nat, log. provided CB is small

in respect of CK.

By Prop. XXIV. the quantity of redundant fluid in the cylinder will bear the same proportion to that in the globe in whatever part the canal meets the cylinder, therefore first I say the redundant fluid in the cylinder will bear a greater proportion to that in the globe than CK CK that of 2CB CB

to nat. log.

For let the canal TR be straight and perpendicular to BL, and let it meet the cylinder in R, the middle point of the line BL, and let it, if produced, meet the axis in S, which will consequently be the middle point of CK; then, if the redundant fluid in the cylinder was spread uniformly on its convex surface, the quantity of redundant fluid therein CK CK would be to that in the globe very nearly as to nat. log. 2CB CB'

For in that case the repulsion of the cylinder on the canal RT would be to the repulsion of the same quantity of redundant fluid collected

2SK SK

CK CK
CB 2CB'

in C very nearly as nat. log. to or as nat. log. to

SR SR

and

the force with which the globe repels the canal in the direction TR is the same with which a quantity of redundant fluid equal to that in the globe placed at S would repel it in the contrary direction.

But there can be no doubt but that almost all the redundant fluid in the cylinder will be collected on its surface, and also will be collected in greater quantity near the ends than near the middle, consequently the repulsion of the cylinder on RT will be less than if the redundant fluid was spread uniformly on its convex surface, and therefore the quantity of redundant fluid in it will bear a greater proportion to that in the globe than it would on that supposition.